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Properties

Label 2-475-19.18-c2-0-4
Degree $2$
Conductor $475$
Sign $0.315 - 0.948i$
Analytic cond. $12.9428$
Root an. cond. $3.59761$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.60i·2-s + 3.60i·3-s − 8.99·4-s + 12.9·6-s + 5·7-s + 18.0i·8-s − 3.99·9-s − 10·11-s − 32.4i·12-s − 3.60i·13-s − 18.0i·14-s + 28.9·16-s − 15·17-s + 14.4i·18-s + (−6 + 18.0i)19-s + ⋯
L(s)  = 1  − 1.80i·2-s + 1.20i·3-s − 2.24·4-s + 2.16·6-s + 0.714·7-s + 2.25i·8-s − 0.444·9-s − 0.909·11-s − 2.70i·12-s − 0.277i·13-s − 1.28i·14-s + 1.81·16-s − 0.882·17-s + 0.801i·18-s + (−0.315 + 0.948i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.315 - 0.948i$
Analytic conductor: \(12.9428\)
Root analytic conductor: \(3.59761\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1),\ 0.315 - 0.948i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5148149377\)
\(L(\frac12)\) \(\approx\) \(0.5148149377\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 + (6 - 18.0i)T \)
good2 \( 1 + 3.60iT - 4T^{2} \)
3 \( 1 - 3.60iT - 9T^{2} \)
7 \( 1 - 5T + 49T^{2} \)
11 \( 1 + 10T + 121T^{2} \)
13 \( 1 + 3.60iT - 169T^{2} \)
17 \( 1 + 15T + 289T^{2} \)
23 \( 1 + 35T + 529T^{2} \)
29 \( 1 - 18.0iT - 841T^{2} \)
31 \( 1 + 36.0iT - 961T^{2} \)
37 \( 1 - 21.6iT - 1.36e3T^{2} \)
41 \( 1 - 36.0iT - 1.68e3T^{2} \)
43 \( 1 - 20T + 1.84e3T^{2} \)
47 \( 1 + 10T + 2.20e3T^{2} \)
53 \( 1 - 75.7iT - 2.80e3T^{2} \)
59 \( 1 - 18.0iT - 3.48e3T^{2} \)
61 \( 1 + 40T + 3.72e3T^{2} \)
67 \( 1 + 39.6iT - 4.48e3T^{2} \)
71 \( 1 - 108. iT - 5.04e3T^{2} \)
73 \( 1 + 105T + 5.32e3T^{2} \)
79 \( 1 + 36.0iT - 6.24e3T^{2} \)
83 \( 1 - 40T + 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + 122. iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.86705634616638408814208592843, −10.24698051286525255142700647459, −9.694887935871901280984487236939, −8.688811452020846393661972312863, −7.86198367237999456135358617414, −5.73695616801405915833295792342, −4.62329161509123409735729978492, −4.13193360861370425539908580404, −2.94804569436370775436398590938, −1.75019018244734823419194157173, 0.20493198409441634789163732938, 2.10095601710012672873327671966, 4.30598937513863677349733840777, 5.22851241826842432970156448305, 6.27463191768634293377222781043, 6.95032335826786165132171790800, 7.76794793911470860274076314771, 8.277385407913304338628207724717, 9.188352421378282561183553718204, 10.50396924974894349061832630969

Graph of the $Z$-function along the critical line